The figure below shows a right triangle:

Question

A right triangle is shown with hypotenuse equal to q units and the length of the legs equal to p units and r units. The angle between the legs having lengths p units and q units is y degrees. The angle between the legs having lengths r units and q units is x degrees.

What is p ÷ r equal to? (4 points)

tan x°

tan y°

sin y°

cos x°

Answer 1

In the right triangle described, we have legs of lengths \( p \) and \( r \), a hypotenuse of length \( q \), and angles \( x \) and \( y \).

According to the definitions of the trigonometric functions for a right triangle, we can relate the sides of the triangle to the angles:

1. For angle \( x \): \[ \tan x = \frac{p}{r} \]

   From this equation, we can rearrange it to express \( \frac{p}{r} \): \[ \frac{p}{r} = \tan x \]

2. For angle \( y \): \[ \tan y = \frac{r}{p} \] Therefore, \[ \frac{r}{p} = \tan y \quad \Rightarrow \quad \frac{p}{r} = \frac{1}{\tan y} = \cot y \]

So, the correct expression for \( \frac{p}{r} \) in terms of the tangent of angle \( x \) is: \[ \frac{p}{r} = \tan x \]

Thus, \( p \div r \) equals:

\[ \boxed{\tan x°} \]